Uniform probabilities maximize every binomial moment of U_j^N and induce opposite radial monotonicity in the PGF of U_j^N on either side of z=1 for the siblings coupon collector.
The Siblings of the Coupon Collector
2 Pith papers cite this work, alongside 4 external citations. Polarity classification is still indexing.
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abstract
The following variant of the collector's problem has attracted considerable attention relatively recently (see, e.g., N. Pintacuda 1980, D. Foata H. Guo-Niu and B. Lass 2001, D. Foata and D. Zeilberger 2003, I. Adler, S. Oren and S. Ross 2003, and S. Ross 2010): There is one main collector who collects coupons. Assume there are $N$ different types of coupons with, in general, unequal occurring probabilities. When the main collector gets a "double", she gives it to her older brother; when this brother gets a "double", he gives it to the next brother, and so on. Hence, when the main collector completes her collection, the album of the $j$-th sibling, $j = 2, 3, \dots$, will still have $U_j^N$ empty spaces. In this article we develop techniques of computing asymptotics of the average $E[U_j^N]$ of $U_j^N$ as $N \rightarrow \infty,$ for a large class of families of coupon probabilities. We also give various illustrative examples.
fields
math.PR 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Proves that uniform coupon probabilities uniquely maximize expected missing types for sibling collectors, that the count is stochastically increasing in N under uniformity, and that normalized counts for fixed siblings converge jointly to the same exponential random variable.
citing papers explorer
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Radial Transform Extremality for the Siblings of the Coupon Collector
Uniform probabilities maximize every binomial moment of U_j^N and induce opposite radial monotonicity in the PGF of U_j^N on either side of z=1 for the siblings coupon collector.
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Extremality and Limit Laws for the Siblings of the Coupon Collector
Proves that uniform coupon probabilities uniquely maximize expected missing types for sibling collectors, that the count is stochastically increasing in N under uniformity, and that normalized counts for fixed siblings converge jointly to the same exponential random variable.